particle properties
space time: angular momentum, Positions
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Particles have properties that seemed tied to some internal structure. This is similar to the color of a ball. The property is an inherent feature of the ball. Its motion (momentum) is tied to its spacetime properties. Internal properties are independent of space time
Intrinsic properties standard
charge, mass (Usually we choose eigenstates of definite mass and charge) [Mass referred to here is the rest mass of the particle.]
Spin seems to bridge the gap. There is some internal structure associated with particles that provides the particle with an intrinsic angular momentum. This behaves in the standard way that angular momentum behaves and must be added to orbital angular momentum. An electron can have its spin reoriented from right to left pointing. This constitutes a change of angular momentum of 1 unit [_{}]. If this were part of the change associated with an atomic transition then the electromagnetic field would need to have the compensating angular momentum change to keep angular momentum conserved.
Rotations are our go to transformations:
_{} There are unitary operators that rotate quantum states.
_{}There are unitary operators that translate states.
Unitary is the complex space analog of orthogonal. Rotations do not change the length of a vector. Unitary transformations do not change the “amount” of quantum states. This can be more formally stated in terms of probability.
Note: If you choose the standard position space
representation for the momentum operator _{} where for simplicity
we consider only one dimension and if we remember the
Quantum numbers
momentum, angular momentum, spin, mass, charge, flavorè(isospin, strangness, charm), electron number, parity, energy, lepton number, baryon number,
Reminder:
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matrix operates to produce new column 
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matrix operates but the result is just a constant times the initial columnè eigenvector of the operator. 
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operator has no inverse. There are various properties that can be ascertained as to the behavior of operators. 
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derivative operates to produce new function 
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derivative operates but the result is just a constant times the initial functionè eigenvector of the operator. 
_{} form of the rotation operator where J is the angular momentum operator.
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Operators don’t commute so we label are states with the values of total and zcomponent.
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_{} Raising and lowering or ladder operators.
These operators produce a new state that has the zcomponent of angular momentum increased/decreased by 1.
The key details concerning structure is that are a subset of operators and associated quantum numbers to label multiplets and there are operators that move you through the multiplet.
We can build new multiplets by combining multiplets. Two spin particles can be combined to for a composite system (hydrogen atom)
_{} use individual quantum numbers to label state
_{} use the total angular momentum and the total zprojection.
Both sets of states span the space.
individual Q# for spin 1/2 
total Q# for spin 1/2 
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Review: [various web sources]
Particles 
